We demonstrate how explicit analytic ruin probabilities can be obtained for claimsize distributions concentrated on a compact interval and constructed arbitrarily from polynomials, exponential functions e(ax) (a is an element of R) and trigonometric functions sin(bx) (x is an element of R), cos(cx) (x is an element of R), by a finite number of additions and multiplications. The method applies also in case of unbounded claims, and then its execution is much simpler than for bounded claims.